Mass and Age of Red Giant Branch Stars Observed with LAMOST and Kepler

MNRAS 475, 3633–3643 (2018) doi:10.1093/mnras/stx3296 Advance Access publication 2017 December 22

Mass and age of red giant branch stars observed with LAMOST and Kepler

Yaqian Wu,1‹ Maosheng Xiang,2‹† Shaolan Bi,1‹ Xiaowei Liu,3 Jie Yu,4,5 Marc Hon,6 Sanjib Sharma,4 Tanda Li,4,5,2 Yang Huang,3† Kang Liu,1 Xianfei Zhang,1 Downloaded from https://academic.oup.com/mnras/article-abstract/475/3/3633/4772882 by Sydney College of Arts user on 17 February 2019 Yaguang Li,1 Zhishuai Ge,1 Zhijia Tian,3† Jinghua Zhang1 and Jianwei Zhang1 1Department of Astronomy, Beijing Normal University, Beijing 100875, P. R. China 2National Astronomical Observatories, Chinese Academy of Sciences, Beijing 100012, P. R. China 3Department of Astronomy, Peking University, Beijing 100871, P. R. China 4School of Physics, University of Sydney, Sydney 2000, NSW, Australia 5Stellar Astrophysics Centre, Department of Physics and Astronomy, Aarhus University, Ny Munkegade 120, DK-8000 Aarhus C, Denmark 6School of Physics, The University of New South Wales, Sydney 2052, NSW, Australia

Accepted 2017 December 19. Received 2017 December 19; in original form 2017 September 25

ABSTRACT Obtaining accurate and precise masses and ages for large numbers of giant stars is of great importance for unraveling the assemblage history of the Galaxy. In this paper, we estimate masses and ages of 6940 red giant branch (RGB) stars with asteroseismic parameters deduced from Kepler photometry and stellar atmospheric parameters derived from LAMOST spectra. The typical uncertainties of mass is a few per cent, and that of age is ∼20 per cent. The sample stars reveal two separate sequences in the age–[α/Fe] relation – a high-α sequence with stars older than ∼8 Gyr and a low-α sequence composed of stars with ages ranging from younger than 1 Gyr to older than 11 Gyr. We further investigate the feasibility of deducing ages and masses directly from LAMOST spectra with a machine learning method based on kernel based principal component analysis, taking a sub-sample of these RGB stars as a training data set. We demonstrate that ages thus derived achieve an accuracy of ∼24 per cent. We also explored the feasibility of estimating ages and masses based on the spectroscopically measured carbon and nitrogen abundances. The results are quite satisfactory and signiﬁcantly improved compared to the previous studies. Key words: asteroseismology – stars: evolution – stars: fundamental parameters.

(HRD) with the theoretical stellar isochrones, utilizing stellar at- 1 INTRODUCTION mospheric parameters yielded by spectroscopy and/or photometry. Accurate age estimates for large numbers of stars are important for a The method has been successfully applied to obtain reasonable age full understanding of the stellar populations and the assemblage his- estimates for a few thousand F/G type stars with high-resolution tory of the Galaxy. Age is one of the key parameters that determine spectroscopy (Edvardsson et al. 1993; Takeda 2007; Haywood et al. the evolutionary state of a star. However, unlike other parameters, 2013; Bergemann et al. 2014)orubvyβ photometry, compled with such as mass and chemical composition, direct age estimate for precise trigonometric parallax data (Nordstrom¨ et al. 2004). Re- a star from observation or fundamental physical law is extremely cently, Wu et al. (2017) used this method to obtain accurate ages for difficult if not impossible (e.g. Soderblom 2010). Often, one has about 200 main-sequence turn-off stars and used the results to check to rely on stellar evolutionary models for stellar age estimation – the accuracy of age estimation for tens of thousand main-sequence usually achieved by comparing the stellar parameters deduced from turn-off and subgiant stars targeted by the LAMOST spectroscopic the observation with the model predictions. surveys (Xiang et al. 2015a, 2017b). However, the method is diffi- A practical way of age estimation for individual field stars is cult to apply to giant stars, as isochrones of giants of different ages to compare their position on the Hertzsprung–Russell diagram are severely crowded together on the HRD, particularly in temperature. On the other hand, giant stars are advantaged probes of the Galactic structure as they can be detected to large distances given E-mail: [email protected] (YW); [email protected] (MX); their high luminosities. Modern large-scale surveys, such as the [email protected] (SB) LAMOST Galactic spectroscopic surveys (Deng et al. 2012;Zhao † LAMOST Fellow. et al. 2012) and the Apache Point Observatory Galactic Evolution

C 2017 The Author(s) Published by Oxford University Press on behalf of the Royal Astronomical Society 3634 Y. Wu et al.

Experiment (APOGEE; Majewski et al. 2010), have collected high- accurate parameters, stellar masses can be inferred to a precision of quality spectra that yield robust stellar atmospheric parameters and ∼7 per cent. About 14 000 stars spectra in the sample of Yu et al. chemical abundances for millions of giant stars. It has become an were targeted by LAMOST by 2016 June. Precise stellar parame- emergent task to develop an effective way that delivers robust age ters and elemental abundances, (e.g. [α/Fe], [C/H], and [N/H]), of estimates for large numbers of giant stars detected by the existing those stars were derived from the LAMOST spectra with the LSP3 and upcoming large spectroscopic surveys. (Xiang et al. 2015b, 2017a). Meanwhile, systematic errors of the Asteroseismologic studies have demonstrated that solar-like os- standard scaling relation were better assessed (White et al. 2011;

cillations excited by convective turbulence in the stellar envelope Guggenberger et al. 2016;Sharmaetal.2016; Viani et al. 2017). Downloaded from https://academic.oup.com/mnras/article-abstract/475/3/3633/4772882 by Sydney College of Arts user on 17 February 2019 are always solar-like stars of 0.8–3.0 M (Aerts, Christensen- With these advantages, it seems that the time is ripe for better mass Dalsgaard & Kurtz 2010; Soderblom 2010; Chaplin & Miglio 2013). and age estimates for the large number of giants available from Both the grid modelling that fits the individual observed frequencies LAMOST surveys. (e.g. Chaplin et al. 2014) and a scaling relation based on global as- In this work, we determine masses and ages for the above teroseismic parameters, ν, the frequency separation of two modes LAMOST–Kepler common red giant branch (RGB) stars. The re- of the same spherical degree and consecutive radial order, and νmax, sults are shown to have a median error of 7 per cent in mass and frequency with the maximal oscillation power (Kjeldsen & Bedding 25 per cent in age estimates. With this LAMOST–Kepler sample, we 1995), have been used to infer mass and radius from the oscillation investigated possible correlations between stellar age and chemical data. Once the precise mass of a giant star has been determined, composition. We further selected a sub-sample of stars with smaller robust age, to an accuracy of about 20–30 per cent (e.g. Soderblom uncertainties in mass and age estimates as a training set to estimate 2010; Gai et al. 2011; Chaplin et al. 2014), then can be estimated by masses and ages directly from the LAMOST spectra with a ma- comparing the inferred mass with the theoretical stellar isochrones chine learning method based on kernel-based principal component of given metallicity, since the main-sequence lifetime of a star is analysis (KPCA). Feasibility of the method is discussed, and the tightly correlated with its mass (e.g. Martig et al. 2016; Ness et al. resultant age and mass estimates are compared with those inferred 2016). Precise photometry from the CoRoT (Baglin 2006), Kepler from an empirical relation based on the C and N abundances. We (Borucki et al. 2010), and K2 (Howell et al. 2014) space missions found that with the current approach, one can derive stellar ages has allowed determinations of seismic parameters for thousands of from the LAMOST spectra with a high precision of 23 per cent. red giant stars (e.g. De Ridder et al. 2009; Hekker et al. 2009, 2011; The paper is organized as follows. Section 2 introduces the Bedding et al. 2010;Mosseretal.2010; Stello et al. 2013, 2015, LAMOST–Kepler sample. Section 3 describes the mass and age es- 2017; Mathur et al. 2016;Yuetal.2016). Many of those stars have timation. Correlations between the age and elemental abundances also reliable atmospheric parameters yielded by spectroscopy, ei- are presented in Section 4. Section 5 presents ages and masses esti- ther from the literature (e.g. Huber et al. 2013; Chaplin et al. 2014; mated directly from the LAMOST spectra with the KPCA method. Pinsonneault et al. 2014), or targeted by the APOGEE (Majewski Section 6 discusses age and mass estimation based on the abundance et al. 2010)ortheLAMOST–Kepler surveys (De Cat et al. 2015), ratio [C/N], followed by conclusions is Section 7. so the ages of these stars can, in principle, be well-determined and adopted as benchmarks for age estimation for large numbers of 2 THE LAMOST–KEPLER SAMPLE giant stars detected in large-scale surveys. Martig et al. (2016) estimated mass and age for 1475 red giants 2.1 Atmospheric and asteroseismic parameters in the APOKASC sample that have atmospheric parameters and elemental abundances including [C/Fe] and [N/Fe] deduced from By 2016 June, the LAMOST–Kepler project (De Cat et al. 2015) APOGEE spectra, as well as seismic parameters derived from Ke- collected more than 180 000 low-resolution (R ∼ 1800) optical pler photometry (Pinsonneault et al. 2014). Based on the fact that spectra (λ 3800–9000 Å) in the Kepler field utilizing the LAMOST values of carbon and nitrogen abundance ratio [C/N] of red giant spectroscopic survey telescope (Cui et al. 2012). Robust stellar stars is tightly correlated with mass as a consequence of the CNO parameters, including radial velocity Vr, effective temperature Teff, cycle and the first-dredge up process, they further constructed a re- surface gravity log g, and metallicity [Fe/H], were delivered from the lation of stellar mass and age as a function of C and N abundances. spectra with the LAMOST Stellar Parameter pipeline (LASP; Wu Using the APOKASC sample as a training set, Ness et al. (2016)es- et al. 2011, Luo et al. 2015). Independently, in addition to the above timated masses and ages for 70 000 APOGEE red giants [including four parameters, interstellar reddening EB − V, absolute magnitudes red clump (RC) stars] with Cannon, a spectroscopic stellar param- MV and MKs , α-element to metal (and iron) abundance ratio [α/M] eters determination pipeline (Ness et al. 2015). Recently, Ho et al. and [α/Fe], as well as carbon and nitrogen abundance [C/H] and (2017) applied the method of Martig et al. (2016) to LAMOST DR2 [N/H], have also been derived with the LAMOST Stellar Parameter and obtained age estimates for 230 000 red giant stars. Nevertheless, Pipeline at Peking University (LSP3; Xiang et al. 2015b, 2017b), we note that due to the relatively large (5 per cent) uncertainties in utilizing spectra processed with a specific flux calibration pipeline the asteroseismic parameters of the APOKASC sample stars, typ- aiming to get better treatment with interstellar extinction of the flux ical uncertainties of stellar masses inferred from the asteroseismic standard stars (Xiang et al. 2015c). Given a spectral signal-to-noise measurements are larger than 0.2 M. As a consequence, the resul- ratio (S/N; 4650 Å) higher than 50, stellar parameters yielded by tant age estimates for the APOKASC sample stars, as well as the LSP3 have typical precision of about 100 K for Teff, 0.1 dex for aforementioned age estimates for the APOGEE and LAMOST gi- log g, 0.3–0.4 mag for MV and MKs, 0.1 dex for [Fe/H], [C/H] and ant stars using the APOKASC sample as a training set, have typical [N/H], and better than 0.05 dex for [α/M] and [α/Fe] (Xiang et al. uncertainties larger than 40 per cent. Better mass and age estimates 2017c). of smaller errors are therefore clearly desired. In this work, we used the global asteroseismic parameters, i.e. Recently, Yu et al. (2017) re-derived asteroseismic parameters, νmax and ν systematically measured by Yu et al. (2017). They including ν and νmax, utilizing the 4-yr Kepler data, with typical used the full-length of four-year Kepler time series and a modified uncertainties smaller than 3 per cent for red giant stars. With these version of the SYD pipeline (Huber et al. 2009) to precisely extract

MNRAS 475, 3633–3643 (2018) Mass and age of RGBs observed with the LAMOST and Kepler 3635 the seismic parameters. Their results are in good agreement with literatures (e.g. Huber et al. 2011; Hekker et al. 2011; Stello et al. 2013; Huber et al. 2014; Mathur et al. 2016;Yuetal.2017), display- ing a median fractional residual of 0.2 per cent in νmax with a scatter of 3.5 per cent, and a median fractional residual of 0.01 per cent in ν with a scatter of 4.2 per cent. Meanwhile, typical uncertainties of the derived seismic param-

eters are smaller than 3 per cent for red giant stars with log g Downloaded from https://academic.oup.com/mnras/article-abstract/475/3/3633/4772882 by Sydney College of Arts user on 17 February 2019 2.0 dex. A cross-identiﬁcation of LAMOST–Kepler stars yields 13 504 LAMOST spectra with S/Ns higher than 20 for 8654 unique stars.

2.2 Selection of the RGB stars Considering that RC stars suffer from significant mass-loss poorly constrained with the current data and the impact of mass-loss on age estimation remains to be investigated in detail, we focus on RGB stars only in this work. To pick out RGB stars, we first select stars with νmax > 120 µHz and νmax < 12 µ Hz, as they are genuine RGB stars. In the regime of 12 µHz <νmax < 120 µHz, RGB, and RC stars are mixed together, we thus use the results of Hon, Stello &Yu(2017), who classified the evolutionary state with a machine learning method utilizing the seismic parameters from Yu et al., as well as other results in the literature that differentiate RGB and RC stars (Bedding et al. 2011; Stello et al. 2013; Mosser et al. 2014; Vrard, Mosser & Samadi 2016;Elsworthetal.2017). Finally, we end up with 6940 RGB stars that have LAMOST spectra. The top panel of Fig. 1 shows the distribution of those RGB sample stars in the HRD. The effective temperatures of the sample stars cover the range of 3500–5500 K and the surface gravities vary from 1.5 to 3.3 dex. The bottom panel of Fig. 1 plots their distributions in the [Fe/H]–[α/Fe] plane. The Figure shows two prominent sequences of stars in the plane of chemical composition, which is corresponding to the thin and thick disc sequence, respectively. Above the line are thick disc stars of high [α/Fe] and low [Fe/H] values, whereas those below are thin disc stars of low [α/Fe] and high Figure 1. Distribution of the RGB sample stars in the Teff–log g (upper) and [Fe/H] values. The distribution resembles those from the previous the [Fe/H]–[α/Fe] (lower) planes. The solid line delineates the demarcation of the thin and thick disc sequences of Haywood et al. (2013). high-resolution spectroscopic studies (e.g. Haywood et al. 2013; Hayden et al. 2015) or predicted by Galactic chemical modelling (e.g. Schonrich¨ & Binney 2009a). Note that there also exists a small number of extremely metal-poor stars with [Fe/H] < −1.0 dex, Combinatio of equations (1) and (2) yields the standard seismic probably belong to the halo population. scaling relation that links the stellar mass and radius with seismic parameters and effective temperature: ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ −4 3 1.5 M ν ν T 3 AGE DETERMINATION = ⎝ ⎠ ⎝ max ⎠ ⎝ eff ⎠ , (3) M ν νmax, Teff, 3.1 Determining masses with seismic scaling relations ⎛ ⎞ ⎛ ⎞⎛ ⎞ Oscillations of solar-like stars are usually described by two global −2 0.5 seismic parameters, the frequency of maximum power, ν , and the R ν νmax Teff max = ⎝ ⎠ ⎝ ⎠⎝ ⎠ . (4) mean large frequency separation, ν. νmax scales with the acoustic R ν νmax, Teff, cutoff frequency (Brown et al. 1991) that depends mainly on surface = = µ gravity and temperature (Kjeldsen & Bedding 1995; Belkacem et al. We adopt solar values Teff, 5777 K, νmax, 3090 Hz and 2011): ν = 135.1 µHz (Huber et al. 2011). It is suggested that un- − − certainty of the standard scaling relation is about 3–5 per cent for ν ∝ gT 1/2 ∝ MR−2T 1/2. (1) max eff eff solar-type main-sequence stars and 10–15 per cent for RGB stars ν is directly related to the travel time of the sound from the centre (Huber et al. 2011). To reduce the systematic errors, there are sev- to the surface of a star, and is therefore sensitive to the mean stellar eral works that attempt to modify the scaling relations (White et al. density (Tassoul 1980; Ulrich 1986; Kjeldsen & Bedding 1995), 2011; Guggenberger et al. 2016;Sharmaetal.2016; Viani et al. 2017). We adopted the modification strategy of Sharma et al. (2016), 1/2 1/2 −3/2 ν ∝ ρ ∝ M R . (2) because it considers in detail the effect on mass, [Fe/H] and Teff

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Figure 2. Distributions of mass (left-hand panel) and age (right-hand panel) estimates, as well as their errors, for the RGB sample stars. The red histogram in the left-hand panel gives the mass error distribution of the APOKASC sample adopted by Martig et al. (2016).

(interpolation over a grid of models), and moreover, the code to do KPCA method from LAMOST spectra. Note that we have cor- this is publicly available. The standard relations then become rected the LSP3 effective temperatures to the scale as given by the ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 3 −4 1.5 colour–metallicity–temperature relation for giant stars of Huang et al. (2015). The Yonsei–Yale (Y2) isochrones (Demarque et al. M = ⎝ νmax ⎠ ⎝ ν ⎠ ⎝ Teff ⎠ , (5) 2004) were adopted because the data base cover a wide range of M νmax, fν ν Teff, age, [Fe/H] and [α/Fe]. The Dartmouth Stellar Evolution Database (DESP; Dotter et al. 2008) and the PAdova and TRieste Stellar ⎛ ⎞⎛ ⎞ ⎛ ⎞ −2 0.5 Evolution Code (PARSEC; Bressan et al. 2012) were also used in R ν ν T = ⎝ max ⎠⎝ ⎠ ⎝ eff ⎠ . (6) order to examine the uncertainty of age estimation induced by the R νmax, fν ν Teff, stellar models. Comparing the results of different isochrones, we found that the age estimated with the DSEP isochrones are com- Here, fν is a modification factor. We derived masses for LAMOST– parable to those estimated with the Y2 isochrones, with a mean Kepler RGB stars with the modified scaling relations, utilizing tem- difference of 6 per cent and a dispersion of 8 per cent. Ages esti- peratures from the LSP3 KPCA method, ν and νmax fromYuetal., mated with the PARSEC isochrones are also consistent well with and fν generated with the ASFGRID code (Sharma et al. 2016). Errors those estimated with the Y2 isochrones. Note that all the three of masses are estimated via propagating errors of ν, νmax and Teff. isochrones did not consider effects from stellar rotation, metal dif- The left-hand panel of Fig. 2 plots distributions of the derived fusion, and magnetic fields, which may have some impact on the age mass estimates and their percentage errors for the RGB sample stars. determination. Most stars have mass between 0.8 and 1.8 M, and the percentage As an example of the age estimation, Fig. 3 plots the posterior errors peak at 7 per cent. There are some stars with relatively large probability distributions as a function of age for three stars of typical (>40 per cent) errors, most of them belong to upper RGB that seis- ages. The existence of prominent and well-defined peaks of the mic parameters with relatively large uncertainties. Also over-plotted distributions suggest that the resultant ages are well constrained. in the Figure is the histogram of mass errors of the APOKASC sam- The robustness of our age estimation benefits to a large extent ple stars adopted by Martig et al. (2016) and Ness et al. (2016)for from the precise mass estimation from seismology. Distributions of age estimation. It shows that the current sample has on average sig- estimated ages and errors for the whole RGB sample are plotted in nificantly smaller mass errors than those of the APOKASC, which the right-hand panel of Fig. 2. The sample covers the whole range of has a median error larger than 0.14 M. However, note that similar possible ages of stars, from close to zero on the young end, up to the to Martig et al. (2016), we ignored mass errors induced by uncer- age of the universe (∼13.8 Gyr; Planck collaboration XIII). There tainties of the scaling relations itself. There are some hints that the are still a small number of stars with unphysical ages, i.e. older than scaling relations may have considerable uncertainties (Gaulme et al. 13.8 Gyr. This is most likely the consequence of parameter errors 2016). in either asteroseismic or atmospheric. Typical relative age errors are 15 per cent–35 per cent, with a median value of 25 per cent. A small fraction (∼5 per cent) of stars exhibit large age errors 3.2 Estimating ages with isochrones (>50 per cent). Again this is mainly due to large uncertainties in By combining masses inferred from the asteroseismic parameters mass estimates and/or in stellar atmospheric parameters. with stellar atmospheric parameters, ages of RGB stars can be es- Sample entries of all parameters deduced for the RGB sample are timated using stellar isochrones. To estimate the stellar age, we listed in Table 1. In addition to the deduced parameters, Teff,logg, adopted a Bayesian method similar to Xiang et al. (2017). The [Fe/H], [α/Fe], [C/N], mass and age, additional information about input stellar parameters are masses and log g inferred from seis- the stars, including the KIC ID, are given. The sample entries of all mic parameters, and Teff,[Fe/H],and[α/Fe] derived with the LSP3 parameters are public available online.

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Figure 3. Posterior probability distributions for age estimation for three example stars of different ages. The vertical and horizontal lines in red denote the mean ages and errors, respectively, whose values are marked in the Figure.

Table 1. Sample entries of deduced parameters of the 3726 uniq RGB sample stars.

Star Teff log g [Fe/H] [α/Fe] [C/N] Mass Age KIC (K) (dex) (dex) (dex) (dex) M (Gyr)

7866696 4291 ± 147 2.54 ± 0.08 −0.02 ± 0.13 0.05 ± 0.06 −0.21 ± 0.12 1.13 ± 0.12 10.46 ± 3.91 7728741 4482 ± 160 3.01 ± 0.16 0.05 ± 0.13 0.04 ± 0.05 −0.19 ± 0.10 1.00 ± 0.08 14.70 ± 3.34 7728945 4945 ± 125 3.05 ± 0.15 −0.23 ± 0.14 0.01 ± 0.05 −0.32 ± 0.10 1.31 ± 0.09 4.03 ± 1.01 10382554 4592 ± 111 2.76 ± 0.09 −0.30 ± 0.09 0.22 ± 0.03 0.02 ± 0.08 1.02 ± 0.06 12.24 ± 3.10 8004863 4843 ± 80 2.95 ± 0.08 −0.15 ± 0.12 0.05 ± 0.03 −0.16 ± 0.08 1.35 ± 0.07 4.18 ± 0.84 7916800 5039 ± 143 3.08 ± 0.19 −0.31 ± 0.15 0.02 ± 0.06 0.03 ± 0.12 1.14 ± 0.08 6.54 ± 1.75 7798102 4638 ± 99 2.54 ± 0.07 −0.48 ± 0.10 0.24 ± 0.03 0.02 ± 0.07 1.06 ± 0.07 9.45 ± 2.46 ...

4 CORRELATIONS AMONG MASS, AGE, more mixed, mostly located in the solar neighbourhood or the outer METALLICITY, AND ABUNDANCE disc. Although the sample of Ho et al. (2017) has also a broad spa- tial distribution, their age estimates are based on the APOKASC Utilizing this sample of RGB stars, we explore possible correlations sample, which, as in our case, contains mainly stars in the among the mass, age, metallicity, and elemental abundances. To inner disc. ensure high data quality, stars with age errors larger than 40 per cent In the [Fe/H]–[C/N] plane, stars with higher [C/N] tend to have or mass errors larger than 15 per cent were discarded, leaving 3726 older ages for a given [Fe/H], likely a consequence of the CNO unique stars in the remaining sample. cycle. The age distribution of stars in the plane however does not Fig. 4 plots the distributions of the median stellar ages and masses follow clear sequences as in the [Fe/H]–[α/Fe] plane, as stars with in the [Fe/H]–[α/Fe] and [Fe/H]–[C/N] planes. From the Figure, it higher [C/N] values also have much larger age dispersions. Stellar shows clear variations of the median age with [Fe/H] and [α/Fe]. masses exhibit clear variations with abundances in both the [Fe/H]– For a given [Fe/H], stars of higher [α/Fe] have older ages. There [α/Fe] and the [Fe/H] – [C/N] planes. Stars with lower values of is an old (>10 Gyr) sequence of stars on the high-α side, span- metallicity, higher [α/Fe] or [C/N] values have generally lower ning continuously in [Fe/H] from a value of −1.0 dex to a su- masses, a consequence of stellar evolution but the target selection persolar metallicity. The trajectory of this old sequence of stats effects may also play a role – the cuts in T and log g used to in the [Fe/H]–[α/Fe] plane agrees well with the highα, thick disc eff select RGB sample stars tend to discard more massive stars of sequence of stars seen in Fig. 1. The thin disc sequence in the abun- older ages. dance plane are dominated by young and intermediate-age stars, Fig. 5 plots the distributions of stars in the age–[α/Fe], age– respectively. The patterns observed here are generally consistent [Fe/H], and age–[C/N] planes. The distribution in the age–[α/Fe] with those of Ho et al. (2017). In particular, fig. 5 of Ho et al. plane exhibits two prominent sequences of different [α/Fe] values. also shows that stars in the low-α tail of the thick disc sequence The high-α sequence is mainly composed of old (>8 Gyr) stars that are as old as their highα, metal-poor counterparts, although this is have an almost flat age–[α/Fe] relation. The low-α sequence con- less clear in their results due to the much larger uncertainties of tains stars with a wide range of ages, from younger than 1 Gyr to their age estimates than ours. The current results however deviate older than 12 Gyr. Stars older than ∼6 Gyr tend to have a flat age– to some extent from those of Xiang et al. (2017) and Haywood [α/Fe] relation, while for younger stars, the [α/Fe] values decrease et al. (2013), who find slightly younger ages of stars in the low-α with decreasing stellar ages. Such a double sequence distribution is tail of the thick disc sequence. The reasons for this discrepancy largely in agreement with the finding of Xiang et al. (2017), but the are not fully understood. We suspect they are either simply due to current sample seems to contain a higher fraction of old (>10 Gyr), the uncertainties in the age estimates, or caused by the possible low-α stars than that of Xiang et al. (2017), and this difference is population effects, as our sample consists mainly stars of the in- clearly responsible for the differences seen in the age distributions ner disc, while the samples of Xiang et al. and Haywood et al. are

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Figure 4. Upper panels: distributions of median stellar ages of [α/Fe]–[Fe/H] (left-hand panel) and [C/N]–[Fe/H] (right-hand panel). Lower panels: distributions of median stellar masses of [α/Fe]–[Fe/H] (left-hand panel) and [C/N]–[Fe/H] (right-hand panel). The colour bar represents the median mass and age of each bin size. A uniﬁed bin size of 0.05 × 0.025 dex is adopted for all the panels.

Figure 5. Age–[α/Fe] (left-hand panel), age–[Fe/H] (middle panel), and age–[C/N] (right-hand panel) relations for the RGB sample stars. of the two samples in the [Fe/H]–[α/Fe] plane as discussed above. some aspect from the finding of some of the previous studies, for The current sample also include 42 young (<5 Gyr) stars of unex- example, Haywood et al. (2013) and Xiang et al. (2017), who find a pectedly high [α/Fe] values (>0.15 dex). Such young, high-α stars tight age–[Fe/H] correlation for stars older than 8 Gyr. Compared to are also found in previous work (e.g. Chiappini et al. 2015;Martig the latter two studies, the current sample exhibits an unexpectedly et al. 2015). Further studies are needed to understand their origins. large fraction of old (>12 Gyr), metal-rich (−0.2 dex) stars. We One possible explanation is that they were formed near the ends suspect that those stars were born in the inner disc. As discussed of the Galactic bar (Chiappini et al. 2015). It is also suggested that above, we believe that the differences are due to the population those stars are actually evolved blue stragglers (Jofreetal.´ 2016). effects. The distribution of stars in the age–[C/N] plane shows that There is no strong correlation between age and [Fe/H]. At any older stars tend to have larger [C/N], in general, suggesting some given age, stars exhibit a wide range of [Fe/H] values. Such a flat correlation between [C/N] ratio and mass and age. However, for a age–[Fe/H] relation is consistent with the previous findings for solar given age, the dispersion of stars in [C/N] is significant. neighbourhood stars (e.g. Nordstrom¨ et al. 2004; Bergemann et al. 2014). It is probably a combination of the consequences of the mix- 5 ESTIMATING AGES AND MASSES FROM ing process that mixes stars born at various positions with different SPECTRA WITH THE KPCA METHOD [Fe/H] (Roskarˇ et al. 2008;Schonrich¨ & Binney 2009a; Loebman et al. 2011) and of the complex star formation and chemical en- In this Section, we explore a machine learning method based on richment history of the Galactic disc. The result however differ in KPCA ( Scholpokf,¨ Smola & Muller¨ 1998) to estimate ages and

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Figure 7. The dispersion of the relative residuals of age for both the training and test groups as a function of N . Red represents test sample, blue Figure 6. Distribution of the LAMOSTC–Kepler training sample in the PC represents training sample. The solid square and triangle represent the results T –log g planes. eff of two groups and leave one out respectively. masses directly from the LAMOST spectra. The method has been an optimal value of the number of principal components, N ,we successfully used to estimate stellar atmospheric parameters (Xiang PC tried various values from 20 to 900, and examined how parameter et al. 2017a). We refer to Xiang et al. for a detailed introduction residuals between the KPCA estimates and the seismic values vary of how to apply the method to LAMOST spectra. Briefly, KPCA with N . Generally, a large number of N generates smaller resid- is a non-linear method to extract principal components from high- PC PC uals between the KPCA estimates and seismic parameters of the dimensional data sets. It works like PCA but the latter is a linear training sample. As N increases from 20 to 900, the dispersion method working in the data space, whereas KPCA works in a fea- PC of the residuals decreases from 26 per cent to 14 per cent for the ture space generated by the kernel function. Here, we adopted the age, and from 9 per cent to 5 per cent for the mass. However, a large Gaussian radial basis function. A multiple linear model between value of N has the risk of over-fitting. An excessively large N the extracted principal components and the target parameters (age, PC PC is also improper as the results become very sensitive to the spec- mass) was constructed with a regression method utilizing the train- tral noises/imperfections, leading to poor robustness for spectra of ing data sets. Parameters of the target spectra were then estimated relatively low S/N. with the model from their principal components. To determine the optimal NPC, we carried out two experiments. One is to divide the sample stars into two groups of equal number, 5.1 The training sample leave-half-out (k-fold-cv), one group is adopted as the training set to estimate ages and masses of the other, test group of stars. Fig. 7 Sample stars with age errors smaller than 40 per cent or mass errors plots the dispersion of the relative residuals of age for both the smaller than 15 per cent, as selected to study the age–metallicity training and test groups as a function of NPC. It shows that as the NPC relation in Section 4, are defined to be the training sample. The increases from 20 to 900, the dispersion of the training sets decreases sample contains 3726 stars, with a median mass error of 7 per cent contiguously from 28 to 12 per cent, whereas the dispersion of the and a median age error of 20 per cent. Although in the current test sets becomes nearly flat when NPC is larger than 100, indicating work, we focus on age and mass estimate for giant stars, we found that the method may suffer from overfitting at such large number of that keeping a sufficient number of sub-giant and dwarf stars in principal components. This also suggests 100 is an optimal value the training sample is necessary in order to reduce the systematic of NPC. Another experiment is the so-called leave-one-out cross- errors caused by the boundary effects of the method (Xiang et al. validation (LOOCV) – the sample stars (of number N) are divided 2017a). We have therefore added to the training sample another into two groups, a training group containing N − 1starsandatest 314 sub-giant and dwarf stars selected from the LAMOST–Kepler group containing one single star. The age and mass of the test star common star data base that have seismic parameters available from are estimated with the KPCA method taking the training group as the catalogue of Chaplin et al. (2014). In total, the training sample the training set. The exercise is repeated N times to estimate the contains 4040 stars with precise seismic mass and age estimates. ages and masses of all the sample stars. Fig. 7 illustrates that the Fig. 6 shows the distribution of stars in the training sample in the method generates slightly smaller dispersion than that of test sample Teff–log g plane. Effective temperatures of the sample stars range of first experiment. This is largely caused by the different size of ∼ from about 4000 to 6500 K, and log g values range from 2.0 to the training sets, as the LOOCV method uses N − 1starswhilethe 4.5 dex. There are few stars of log g < 2.0 dex, probably because k-fold-cv method uses N/2 stars. For safety, we adopt a value of these stars have relatively large uncertainties of seismic parameters. 100 for NPC in this work. Fig. 8 plots the residuals of ages and masses deduced from spectral fitting with the KPCA method as compared to the seismic values 5.2 KPCA results as a function of metallicity for the 4040 training sample stars. The Similar to Xiang et al. (2017a), we adopted the 3900– Figure shows no significant biases of age and mass estimates for 5500 Å spectral segment for age and mass determination. To find [Fe/H] values down to about −1.5 dex.

MNRAS 475, 3633–3643 (2018) 3640 Y. Wu et al.

6 ESTIMATING AGES AND MASSES BASED ON THE CARBON AND NITROGEN ABUNDANCES The observed C and N abundance ratio [C/N] of a RGB star depends on its mass as a consequence of the ﬁrst dredge-up process, that brings the inner material processed by the CNO cycle out to the stellar surface. This is because [C/N] ratio in the core, as well as the depth of the convective mixing that drives the dredge-up process, Downloaded from https://academic.oup.com/mnras/article-abstract/475/3/3633/4772882 by Sydney College of Arts user on 17 February 2019 depends sensitively on the stellar mass (Masseron & Gilmore 2015). The [C/N] is thus a good indicator of the mass and age of a RGB star (e.g. Masseron & Gilmore 2015; Martig et al. 2016). Following Martig et al. (2016), we estimated ages and masses based on the [C/H] and [N/H] abundances utilizing a polynomial regression method. We assumed a quadratic function between the mass (age) and the stellar parameters. Two sets of stellar parameters were employed, one includes Teff,logg, [Fe/H], [α/Fe], [C/N], and Figure 8. Variations of the regression residuals of age and mass estimates [C/Fe], the other includes Teff,logg, [Fe/H], [C/N], and [C/Fe]: as a functions of [Fe/H]. Mass (Age) = f (Teff , log g,[Fe/H], [α/Fe], [C/N], [C/Fe]), (7)

Mass (Age) = f (Teff , log g,[Fe/H], [C/N], [C/Fe]). (8) Least-square fitting algorithm is adopted to determine the coeffi- To further examine the feasibility of the method, we plot the me- cients of the quadratic function, utilizing a sample of 3726 stars dian residuals of ages and masses as derived with spectral fitting the with age errors smaller than 40 per cent and mass errors smaller seismic values in the Teff–log g plane in Fig. 9. In general, the Fig- than 15 per cent, as defined in Section 4. Coefficients of the fits ure exhibits rather homogeneous distribution for both age and mass are listed in Tables 2 and 3 for the two sets of stellar parameters, residuals across the Teff–logg plane except for some regions near the respectively. boundaries. For sub-giant stars of Teff ∼ 5200 K and logg ∼ 3.5 dex, Fig. 10 plots a comparison of masses deduced with the fitting the KPCA method may yielded ages systematically higher than the formulae equations (7) and (8) and the seismic values. This figure seismic values by ∼3 Gyr, and masses lower than the seismic values shows good agreement, with a dispersion of the residuals of only by ∼0.2 dex. The reasons are not fully understood. One possibility 7 per cent and 8 per cent for the two parameter sets with and with- is a defect in the KPCA method due to small number of sub-giant out [α/Fe]. The results are much more precise than a precision of stars in the training sample. 20 per cent obtained by Martig et al. (2015). Near either the lower The method have been applied to the whole spectra set of the or higher mass end, there are the some visible differences. They are LAMOST Galactic surveys, yielding ages and masses for hundreds likely to be mainly caused by the random errors in seismic mass es- of thousands of RGB stars. A careful analysis of the huge data set, timates, although some uncertainties introduced by the poor fitting as well as a further calibration of age estimates with member stars near the mass boundaries may also contribute some of the differ- of open clusters, will be presented in a separate work. ences. A similar comparison for age estimates is shown in Fig. 11.

Figure 9. Distributions of residuals of ages (left-hand panel) and masses (right-hand panel) estimated with KPCA method as compared to the seismic values in the Teff–log g planes.

MNRAS 475, 3633–3643 (2018) Mass and age of RGBs observed with the LAMOST and Kepler 3641

Table 2. Best-ﬁtting coefﬁcients for mass estimation based on the carbon and nitrogen abundances.

(a) Best-ﬁtting coefﬁcients for mass estimation as a quadratic function of Teff,logg, [Fe/H], [α/Fe], [C/N], and [C/Fe] [see equation 7]

1logTeff log g [Fe/H] [α/Fe] [C/N] [C/Fe] 1 1138.72 −625.59 − 2.66 9.97 37.69 37.55 4.22 log Teff 86.12 1.17 − 3.1 − 10.15 − 11.25 − 1.49 log g − 0.33 0.54 − 0.14 1.07 0.43

[Fe/H] − 0.24 − 0.65 − 0.5 − 0.34 Downloaded from https://academic.oup.com/mnras/article-abstract/475/3/3633/4772882 by Sydney College of Arts user on 17 February 2019 [α/Fe] − 0.15 1.52 1.03 [C/N] 0.29 0.05 [C/Fe] − 0.25 Notes. (b) Best-ﬁtting coefﬁcients for mass estimation as a quadratic function of Teff,logg, [Fe/H], [C/N], and [C/Fe] [see equation 8] 1logTeff log g [Fe/H] [C/N] [C/Fe] 1 109.11 − 30.84 − 50.28 21.22 34.49 92.02 log Teff 0.06 14.65 − 6.36 − 10.19 − 26.39 log g − 0.69 0.85 0.81 1.72 [Fe/H] − 0.39 − 0.38 − 1.35 [C/N] 0.50 − 0.01 [C/Fe] − 1.04

Table 3. Best-ﬁttingcoefﬁcients for age estimation based on the carbon and nitrogen abundances.

(a) Best-ﬁtting coefﬁcients for age estimation as a quadratic function of Teff,logg, [Fe/H], [α/Fe], [C/N], and [C/Fe] [seen equation 7] 1logTeff log g [Fe/H] [α/Fe] [C/N] [C/Fe] 1 − 9749.25 5842.99 − 109.99 603.26 607.98 1039.59 262.19 log Teff − 789.07 27.34 − 167.04 − 181.39 − 282.58 − 73.01 log g 2.71 3.85 20.1 3.14 2.28 [Fe/H] − 2.76 − 32.43 − 5.15 − 3.93 [α/Fe] 7.93 − 16.39 − 14.46 [C/N] − 2.88 − 1.01 [C/Fe] − 4.69

Notes. (b) Best-ﬁtting coefﬁcients for age estimation as a quadratic function of Teff,logg, [Fe/H], [C/N], and [C/Fe] [seen equation 8]

1logTeff log g [Fe/H] [C/N] [C/Fe] 1 − 10318 6017.42 − 184.05 537.77 1236.82 287.04 log Teff − 873.87 50.84 − 151.71 − 340.27 − 82.19 log g 0.83 5.84 8.45 5.58 [Fe/H] 0.40 − 9.79 − 1.44 [C/N] − 2.92 − 1.07 [C/Fe] − 5.11

The residuals have a dispersion of 25 per cent and 26 per cent for the 3726 RGB stars with small age and mass uncertainties, we have two parameter sets with and without [α/Fe]. However, some sys- investigated possible correlations among age, [Fe/H], [α/Fe], and tematic patterns in the residuals are seen as a function of age. Ages [C/N]. The results show that the age–[α/Fe] relation exhibits two estimated with the carbon and nitrogen abundances are larger than separate sequences, a flat, high-α sequence dominated by stars older the seismic values by ∼10 per cent for stars younger than 5Gyr, than ∼8 Gyr, and a low-α sequence composed of stars with a wide and lower than the seismic values by ∼10 per cent for stars older range of ages, from younger than 1 Gyr to older than 13 Gyr. At than 10 Gyr, likely caused by the inadequacy of the regression any given age, stars exhibit a broad [Fe/H] distribution, leading to method. Figs 10 and 11 also show that including [α/Fe] in the fit- a weak age–[Fe/H] correlation. Particularly, there is a considerable ting does not significantly improve the precision of mass and age fraction of metal-rich (−0.2 dex) stars of very old ages (>10 Gyr). estimation. This is probably because [C/Fe] correlates well with Taking the sample stars as a training data set, we show that precise [α/Fe]. ages and masses can be estimated directly from the LAMOST spectra with a KPCA based machine-learning method. Typical uncertainties of masses and ages thus estimated are comparable to those of the seismic estimates. The method can be applied to the whole 7 CONCLUSION data base of the LAMOST Galactic Spectroscopic Surveys, yielding We have estimated masses and ages for 6940 RGB stars observed by robust age and mass estimates for hundreds of thousands of RGB the LAMOST–Kepler project that have accurate asteroseismic pa- stars. We have also explored the feasibility of estimating masses and rameters. Typical uncertainty is ∼7 per cent for the mass estimates, ages from the stellar atmospheric parameters, including the carbon and ∼25 per cent for the age estimates. Utilizing a subsample of and nitrogen abundances, and found that the method is capable of

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Figure 10. Comparison of masses deduced based on the carbon and nitrogen abundances using ﬁts equation (7) (left-hand panel) and equation (8) (right-hand panel). The bottom panels show the residuals, with the resistant estimate of the standard deviation marked in red.

Figure 11. Same as Fig. 10, but for age estimation. yielding masses with a uncertainty of better than 10 per cent, as well This work is also supported by National Key Basic Research Pro- as age with a uncertainty of better than 23 per cent. gram of China 2014CB845700 and Joint Funds of the National Natural Science Foundation of China (Grant No. U1531244). MX and YH acknowledge supports from NSFC Grant No. 11703035. ACKNOWLEDGEMENTS The LAMOST FELLOWSHIP is supported by Special Funding This work is supported by the Joint Research Fund in Astron- for Advanced Users, budgeted and administrated by Center for omy (U1631236) under cooperative agreement between the Na- Astronomical Mega-Science, Chinese Academy of Sciences tional Natural Science Foundation of China (NSFC) and Chi- (CAMS). TL acknowledges funding from an Australian Research nese Academy of Sciences (CAS), and grants 11273007 and Council DP grant DP150104667, the Danish National Research 10933002 from the National Natural Science Foundation of China, Foundation (Grant DNRF106), grants 11503039 and 11427901 and the Fundamental Research Funds for the Central Universi- from the National Natural Science Foundation of China. Gu- ties and Youth Scholars Program of Beijing Normal University. oshoujing Telescope (the Large Sky Area Multi-Object Fibre

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